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The Lindelöf property and fragmentability


Authors: B. Cascales, I. Namioka and G. Vera
Journal: Proc. Amer. Math. Soc. 128 (2000), 3301-3309
MSC (2000): Primary 46A50, 46B22; Secondary 54C35
DOI: https://doi.org/10.1090/S0002-9939-00-05480-0
Published electronically: April 28, 2000
MathSciNet review: 1695167
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Abstract: Let $K$ be a compact Hausdorff space and $C(K)$ the space of continuous real functions on $K$. In this paper we prove that any $t_{p}(K)$-Lindelöf subset of $C(K)$ which is compact for the topology $t_{p}(D)$ of pointwise convergence on a dense subset $D\subset K$ is norm fragmented; i.e., each non-empty subset of it contains a non-empty $t_{p}(D)$-relatively open subset of small supremum norm diameter. Several applications are given.


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Additional Information

B. Cascales
Affiliation: Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30.100 Espinardo, Murcia, Spain
Email: beca@fcu.um.es

I. Namioka
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195–4350
Email: namioka@math.washington.edu

G. Vera
Affiliation: Departamento de Matemáticas, Facultad de Matemáticas, Universidad de Murcia, 30.100 Espinardo, Murcia, Spain
Email: gvb@fcu.um.es

DOI: https://doi.org/10.1090/S0002-9939-00-05480-0
Keywords: Pointwise compactness, Radon-Nikod\'ym compact spaces, fragmentability
Received by editor(s): July 20, 1998
Received by editor(s) in revised form: December 20, 1998
Published electronically: April 28, 2000
Additional Notes: The first and third authors were partially supported by research grant DGES PB 95–1025.
Communicated by: Dale Alspach
Article copyright: © Copyright 2000 American Mathematical Society

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